San José State University

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Thayer Watkins
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The Physical Impossibilities
of Point Particles

Charged particles like the electron and the proton consist of two parts: a core and a field. The core consists of the distribution of charge over a limited range. Here is the empirically determined charge distributions for protons and neutrons.

As can be seen above, a neutron does have a charges distribution even though it has zero net charge and no field.

The Energy of the Field
of a Charged Particle

Consider a stationary particle with charge Q. The electrical field intensity E at a distance r from the center of that particle is given by

E(r) = (1/(4πε0))Q/r²

where ε0 is a constant called the permitivity of space.

For a particle in empty space the energy density U is

U = ½ε0E² = (1/(32π²ε0))Q²/r4

The energy in a spherical shell of radius r and thickness dr is equal to 4πr²(Q²/(32π²r4) which reduces to Q²/(8πε0r²). The integration of these terms from R to ∞ gives a total energy TR

TR = ∫RU(4πr²)dr = Q²/(8πε0R)

If R→0, then TR → ∞ no matter how small is the magnitude of the charge so long as it is nonzero. Thus a charged point particle, if such existed, would have infinite energy.

The Absurdities of a Particle with Infinite Energy

The absurdityof the notion of a single electron having more energy than all of kinetic energy of all particles in all of the galaxies of the Universe need no further elaboration.

It is worth noting that there can be no conservation of energy if there is an infinite source of energy that can be drawn from.

Since a particle and its antiparticle can annihilate each other and their fields, it follows that such annihilation of point particles should produce photons of infinite energy. The mutual annihilation of an electron and a positron produces gamma ray(s) of finite energy. Therefore electrons and positrons cannot be point particles.

How Physical Evidence of the
Pointedness of Particles Can Arise

There is a wonderful theorem in mathematical physics that says that the effect of a spherical charge uniformly distributed is the same as if the charge were concentrated at the center of the sphere. That is for points outside of sphere of the charge; inside that sphere the effect is zero. Therefore the deflection of probe particles impinging upon a spherical shell of charge is the same as if the charge of the shell were concentrated at its center provided that the probe particles do not have enough energy to penetrate within the shell. The effect on probe particles within the shell would be called asymptotic freedom. Thus evidence of a particle having the same effect over a limited range of energies as a point particle is not evidence of the existence of point particles.

The Consequence of the Assummption
in Quantum Field Theories of Fields which
which are those of Point Particles

Quantum Field Theories are plagued with infinities. Various devices such as renormalization group theory were developed to successfully eliminate those infinities. The interesting intellectual question is whether those infinitinities arise because the fields assumed in QFT are the same as those of point particles. But that is a topic for a different webpage.

(To be continued.)


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